Method for operating a linear motor

ABSTRACT

To control movement of a transport unit of a linear motor, a quality functional J(SG) with quality terms JT k (SG) is used as a function of manipulated variables (SG) of active drive coils. The quality functional J(SG) controlling movement of the transport unit along the stator is optimized with regard to the manipulated variable (SG) to determine optimal manipulated variables (SG opt ) for the relevant time step of the control of movement, active drive coils are energized according to the determined optimal manipulated variables (SG opt ), and at least two movement phases are provided during the movement of the transport unit along the stator. In the at least two movement phases, different quality functionals J(SG) are used for determining the optimal manipulated variables (SG opt ), the different quality functionals J(SG) differing by the number k of the quality terms JT k (SG) used and/or by the quality terms JT k (SG) and/or by the weighting Factors k k .

The present invention relates to a method for operating a linear motorcomprising a stator, on which a plurality of drive coils is arranged,and comprising a transport unit, which is moved along the stator and onwhich a plurality of drive magnets is arranged, wherein anelectromagnetic field, which interacts with the drive magnets of thetransport unit for moving the transport unit, is generated by energizingdrive coils in the region of the drive magnets of the transport unit.

In a linear motor, a primary part is provided and a secondary part(rotor) is provided which is arranged so as to be movable relative tothe primary part. Drive coils are arranged on the primary part, anddrive magnets are arranged on the secondary part, or vice versa. Thedrive magnets are designed as permanent magnets, electrical coils, orshort-circuit windings. The drive coils are electrical coils that areenergized to generate an electromagnetic field. Due to the interactionof the (electro)magnetic fields of the drive magnets and the drivecoils, forces act on the secondary part, which forces move the secondarypart relative to the primary part. The linear motor can be designed, forexample, as a synchronous machine or as an asynchronous machine. Thelinear motor may also be designed as a planar linear motor (also calledplanar motor). The drive coils of the linear motor are either arrangedalong a movement direction or, in the case of a planar motor, along twomovement directions, i.e., in a movement plane. The secondary part canbe moved along this one movement direction or can be moved freely in themovement plane in the two movement directions. A distinction can also bemade between short-stator linear motors and long-stator linear motors,the secondary part being shorter than the primary part in thelong-stator linear motor and the primary part being shorter than thesecondary part in the short-stator linear motor. In the planar motor,the primary part is usually larger than the secondary part.

Linear motors are used in electromagnetic transportation systems inwhich a transport unit is moved in order to perform a transport task.The transport unit can be designed as a secondary part or a primarypart. Usually, such an electromagnetic transport system is designed inthe form of a long-stator linear motor or planar motor.

In a long-stator linear motor, a plurality of electrical drive coils arearranged next to one another in a stationary manner along a stator. Anumber of drive magnets, either as permanent magnets or as an electricalcoil or a short-circuit winding, which are separated by an air gap fromthe drive coils and which interact with the drive coils, are arranged ona transport unit. Due to the interaction of the (electro)magnetic fieldsof the drive magnets and the drive coils, forces act on the transportunit, which forces move the transport unit along the stator. In a planarmotor, the drive coils are arranged in a plane on the stator. Likewise,there are drive magnets which are arranged in a plane on the transportunit.

Often, a specific number of drive coils are arranged on individualstator segments. The stator segments can also have different geometries,such as straight lines, curves, switches etc. The stator segments canthen be assembled to form the desired stator by arranging them in a row.However, the use of such stator segments and the number of such statorsegments is irrelevant to the linear motor according to the inventionand so are the number and arrangement of the drive coils on a statorsegment.

By energizing the drive coils in the region of the drive magnets of thetransport unit, a magnetic field can be generated which interacts withthe magnetic field of the drive magnets in order to generate apropulsive force (in the movement direction along the stator or, in thecase of a planar motor, in the movement direction in a movement planealong the stator) and/or a normal force (in the direction transverse tothe movement direction) acting on the transport unit. The resultingforces can be influenced by controlling the individual drive coils inorder to control the generated magnetic flux. As a result, the transportunit can be moved in the desired manner along the transport route (whichcomprises a transport plane in the case of a planar motor) bycontrolling the drive coils to generate a magnetic field that is movedin the movement direction. It is also possible to arrange, along thetransport route, a plurality of transport units the movements of whichcan be controlled individually and independently of one another byenergizing the drive coils, generally by applying an electrical voltage,interacting with a respective transport unit.

Examples of long stator linear motors can be found in WO 2013/143783 A1U.S. Pat. No. 6,876,107 B2, US 2013/0074724 A1 or WO 2004/103792 A1. Forexample, U.S. Pat. No. 9,202,719 B2 discloses the basic structure andmode of operation of a planar motor.

In order to control the movement of a transport unit, a do-coordinatesystem moving along with the transport unit is usually used, analogouslyto rotative motors. In the case of a planar motor also separately in thetwo movement directions. A drive current required for the desiredmovement is then calculated in the do-coordinate system, with a currentcomponent in the movement direction (often also referred to asq-component) and possibly a current component in the normal direction(i.e., transverse to the movement direction, often also referred to asd-component). The q-component is responsible for the generation of thepropulsive force and the d-components are responsible for the generationof a normal force transverse to the propulsive force. However, no activenormal force (i.e., by energizing the drive coils) is generally requiredfor normal movement along the stator. The drive current in thedq-coordinate system is then converted, analogously to the rotativemotor, with an inverse Park transformation into the coil currents of thedrive coils involved in the generation of the forces, which coilcurrents are then generated by applying a corresponding voltage to thedrive coils by means of a power electronics system. This is repeatedwith a time step specified for the control, typically in the 1/10millisecond range. Due to the movement of the transport unit, differentdrive coils will also be involved during the movement. This control isindependent of whether forces are generated on both sides (as viewed inthe movement direction of the transport unit) of the transport unit, oronly on one side.

Such a control on the basis of a do-model of the long-stator linearmotor is described, for example, in Khong, P. C., et al., “Magneticguidance of the mover in a long-primary linear motor” IEEE Transactionson Industry Applications, Vol. 47, No. 3, May/June 2011, pp. 1319-1327.Khong describes a long-stator linear motor with drive coils arranged onboth sides, as viewed in the movement direction, and the normal forceson the two sides are used to center the transport unit in the center forguidance.

EP 3 385 110 A1 describes a long-stator linear motor in which thepropulsive force and the normal force are controlled independently ofone another in order to be able to adapt the normal force to theconditions of the transport route. In EP 3 109 998 B1 the normal forceor a torque is used for the control of the movement of the transportunit in a switch.

For the control of the forces and/or torques acting on the transportunit, a controller, often also a controller cascade consisting ofseveral successive controllers, is used to compensate for a deviationbetween a specified setpoint, for example a setpoint position orsetpoint speed, and an actual variable, for example an actual positionor actual speed. Often, a plurality of controllers are also providedalong the stator, for example one controller per stator segment (asdescribed in EP 3 422 558 A1). On the basis of an implemented controlrule, the controller, often a PI controller or PID controller,determines a manipulated variable, for example electrical voltages to beapplied to the drive coils, which manipulated variable is thenimplemented on the linear motor. For example, the manipulated variableis implemented for the drive coils by a power electronics system, i.e.,generated and applied to the drive coils. A conventional controllercomprises controller parameters which must be adapted to the respectivecontrolled system, here stator with drive coils and transport unit withdrive magnets and the magnetic coupling thereof, in order to set thedesired control behavior, such as control errors, overshooting,stability, response behavior, etc. The controller must therefore beparametrized, which means that corresponding values must be assigned tothe controller parameters. These controller parameters are usuallyfixedly specified and are not changed during operation.

EP 3 251 986 A1 describes the parametrization of the controllerparameters for a long-stator linear motor. In this case, it can also beprovided to use different controller parameter sets for differenttransport units or for different locations of the transport route inorder to be able to adapt the control behavior to different transportunits in the best possible manner, for example on the basis of differentloading, wear state, design etc. For parametrization, the controlledsystem is excited with an excitation signal superimposed on themanipulated variables, and the response of the controlled system to theexcitation in the form of the frequency response is evaluated in orderto derive the controller parameters therefrom. On the one hand, this iscomplicated and can, on the other hand, be done only at a specificlocation of the transport route and for a specific transport unit.Although the control can be improved in this approach, it is stillinflexible.

It is an object of the present invention to improve the control of themovement of a transport unit of a linear motor, in particular to make itmore flexible.

This object is achieved with the features of independent claim 1.

For the most accurate control of the movement, the quality functionalcontains a quality term, which evaluates a deviation of a specifiedsetpoint variable of the control from an actual variable of the controlwhich is dependent on the manipulated variable. This allows a highcontrol quality of the control of the movement of the transport unit. Byusing different quality functionals in different movement phases of themovement of the transport unit, the control can be flexibly adapted tothe respective movement and different control objectives can be pursuedin the movement phases. The movement phases can be configured flexiblyalong the stator, as can the quality functional in the movement phases.This does not change the basic sequence of the control of the movementof the transport unit since only different quality functionals areoptimized.

The optimization-based control of the movement of the transport unitallows both on-line optimization and off-line optimization of thequality functional. This further increases the flexibility of thecontrol of the movement.

In the case of off-line optimization, it is in particular advantageousto create movement characteristic maps for different quality functionalsin advance, which movement characteristic maps map a movement setpointvariable, in particular a driving force, and a position of the transportunit relative to the active drive coils on the optimal manipulatedvariables. The optimal manipulated variables for the control of themovement of the transport unit thus must only be read on-line from thecorrect movement characteristic map.

Preferably, the quality functional contains a quality term, whichevaluates the electrical power required for the movement of thetransport unit. This makes it possible to provide a movement phase inwhich the movement of the transport unit takes place with as littleelectrical power loss as possible. This allows an energy-efficientoperation of the linear motor.

If the quality functional contains a quality term that evaluates a sumof the manipulated variables, the thermal load on the components of thepower electronics system for implementing the manipulated variables, andthe heating of the stator can be reduced.

Advantageously, a mathematical model of the linear motor is used, withwhich a variable of a quality term can be determined or which can beused for a constraint of the optimization. With the model, specificvariables or quantities can be calculated instead of measuring them inthe linear motor. With the model, the state variables of the linearmotor and also all required output variables can also be determined.When a constraint is used, it is ensured that the physical laws of thelinear motor are taken into account, which can improve the controlquality in general.

Particularly advantageously, the linear motor is mathematically modeledwith a reluctance network, i.e., with system equations that are derivedfrom the reluctance network and reflect the physical laws. A high systemquality (accuracy) in comparison to model complexity can be achievedwith a reluctance model.

The present invention is described in greater detail below withreference to FIGS. 1 to 9 , which show schematic and non-limitingadvantageous embodiments of the invention by way of example. Thefollowing are shown:

FIGS. 1 and 2 a possible structure of a long-stator linear motorcomprising a stator with teeth with drive coils and comprising atransport unit with drive magnets,

FIGS. 3 and 4 a reluctance network of the stator,

FIG. 5 a reluctance network of the rotor and the coupling thereof to thestator,

FIG. 6 a control of the movement of a transport unit by optimizing aquality functional,

FIGS. 7 and 8 possible embodiments of the control of the movement of thetransport unit, and

FIG. 9 an example of a movement characteristic map.

The invention is described using the example of a long-stator linearmotor as an embodiment of a linear motor but can also be appliedanalogously to other embodiments of a linear motor, such as a planarmotor or a short-stator linear motor.

FIG. 1 shows, by way of example, a section of a long-stator linear motor1 in a very simplified manner. The long-stator linear motor 1 consistsof a stator 2 and at least one transport unit 3, which can be movedalong the stator 2 in the movement direction x. In the exemplaryembodiment shown, the stator 2 is the primary part of the linear motorand the transport unit 3 is the secondary part. Usually, a plurality oftransport units 3 (also different transport units) are movedsimultaneously and independently of one another along the stator 2. Thestructural design and the geometry of the stator 2 and of the transportunit 3 do not play a role in the invention but naturally influence thestructure of the model of the linear motor 1 described below. How thetransport unit 3 is guided and held along the stator 2 is equallyunimportant to the invention.

A plurality of drive coils 4 are arranged next to one another along thestator 2 in the movement direction x. A drive coil 4 is usually arrangedon a stator tooth 5 made of a magnetically highly permeable material,such as iron, which teeth are connected to one another by a stator yoke6. Between two stator teeth 5 with a drive coil 4, a secondary tooth 7without a drive coil 4 may also be provided. However, the secondaryteeth 7 may also be omitted. A drive magnet arrangement 8 with a numberof drive magnets 9 is arranged on the transport unit 3. Drive magnets 9of the drive magnet arrangement 8 arranged next to one another in themovement direction x are usually oppositely polarized. The drive magnetarrangement 8 faces the drive coils 4 and is separated therefrom by anair gap 14. The maintenance of the air gap 14 is generally ensured bythe usually mechanical and/or magnetic guidance (not shown) of thetransport unit 3 on the stator 2.

As already mentioned, the drive coils 4 in a planar motor would bearranged in a movement plane of the stator 2. The drive magnets 9 wouldlikewise be arranged in a plane. In the case of a planar motor, there isusually no mechanical guidance of the transport unit 3, but thetransport unit 3 is usually held magnetically in suspension.

For the movement of the transport unit 3, a number of drive coils 4 inthe region of the drive magnet arrangement 8 of the transport unit 3interact with the drive magnets 9 of the drive magnet arrangement 8. Thenumber of drive coils 4 used for the movement of the transport unit 3can be assumed to be known and can be defined but can also be dependenton the position of the transport unit 3 relative to the stator 2 and canhowever also be defined depending on the optimization described below.The number of drive coils 4 can thus also arise as a result of theoptimization, the optimization thus determining how many drive coils 4in the environment of the transport unit 3 are to be energized and how.These energized drive coils 4, hereinafter also referred to as activedrive coils, are controlled by a control unit 10. Which of the drivecoils 4 are the active drive coils naturally changes during the movementof the transport unit 3 relative to the stator 2.

However, it should be noted that all drive coils 4 in the region of thedrive magnet arrangement 8 do not necessarily always have to be activedrive coils. Situations are conceivable in which only certain drivecoils 4 in the region of the drive magnet arrangement 8 are used asactive drive coils. For example, a drive coil 4 may have failed and thuscannot be used for the movement of the transport unit 3. Nevertheless,the transport unit 3 can be moved with the other drive coils 4 in theregion of the drive magnet arrangement 8 as active drive coils.

In order to avoid a failure of the long-stator linear motor 1 in theevent of failure of a drive coil 4, provision may be made for thecontrol unit 10 to simply ignore the failed drive coil 4. However, thiscan lead to a certain impairment of the movement of the transport unit3, which may however be negligible in certain situations, for example insections of the movement without accuracy requirements on the movementof the transport unit 3. The control unit 10 may however also compensatefor the failure of a drive coils 4 with the remaining active drive coilsin order to avoid the failure of the long-stator linear motor 1. Forthis purpose, it would merely be necessary to inform the control unit 10which drive coil 4 has failed. This could also be very simply done inthe context of the present invention, as explained below.

In order to establish a failure of a drive coils 4, a coil current or acoil voltage could be applied thereto. It could then be established bymeasuring a coil voltage or a coil current whether a drive coil 4 isshort-circuited or open-circuited, which corresponds to a failurebecause the drive coil can no longer be actuated. This could occurcontinuously during operation of the long-stator linear motor 1, evenduring the control of the movement of the transport unit 3.

Drive coils 4 can also be arranged on both sides of the transport unit 3as viewed in the movement direction x. In this case, the transport unit3 can likewise have a drive magnet arrangement 8 on both sides as viewedin the movement direction x. In such a structure, forces acting on thetransport unit 3 can thus be generated on both sides of the transportunit 3. However, this does not change the control according to theinvention as described below.

The control unit 10 uses a movement setpoint variable BS, which isspecified, for example, by a system control unit 11, for example via adata communication bus as indicated in FIG. 1 . The system control unit11 controls and monitors, for example, the simultaneous movement ofseveral transport units 3 on the stator 2. In each time step of thecontrol, usually in the 1/10 millisecond range, a manipulated variableSG for a number of or each active drive coil 4 involved in the movementof the transport unit 3 is determined by the control unit 10 on thebasis of the movement setpoint variable BS with a control implementedthereon (for example in the form of control software running onprocessor-based hardware). However, a control unit 10 for each drivecoil 4 may also be provided. The manipulated variables SG of the activedrive coils 4 are usually not the same.

The manipulated variable SG may be a coil current i_(c) or a coilvoltage v_(c) with which the drive coil 4 is energized. The coil currenti_(c) or the coil voltage v_(c) can be generated for each involved drivecoil 4 and applied to the drive coils 4 by a power electronics system(not shown), for example as described in EP3 249 803 A1 Usually, themanipulated variable SG is a coil voltage v_(c), which is applied to theactive drive coils 4.

In the case of a coil current i_(c) as the manipulated variable SG aconversion into a coil voltage v_(c) can also be provided, as explainedbelow, or a current controller that adjusts the coil current i_(c) canbe provided. This can take place in the control unit 10 or also in thepower electronics system.

The movement setpoint variable BS can be specified for each time step ofthe control, for example a setpoint position of the transport unit 3along the stator 2, or a setpoint speed or also a setpoint force.However, the movement setpoint variable BS can also be determined in thecontrol unit 10 itself, for example on the basis of a planned movement(e.g., as the course of the position or speed of the transport unit 3over time or distance travelled) along the stator 2. This plannedmovement can be specified to the control unit 10 but can also becalculated in the control unit 10, for example in order to reach aspecified setpoint position along the stator 2.

In order to control the movement of the transport unit 3, it may also beprovided to determine an actual variable of the movement, for example anactual position, of the transport unit 3 along the stator 2, for exampleon the basis of known position sensors 13 (indicated in FIG. 1 ), whichare arranged along the stator 2.

Usually, the necessary driving forces and/or driving torques to beimpressed on the transport unit 3 are determined from the movementsetpoint variable BS in an upstream movement controller for thetransport unit 3 in each time step of the control so that the transportunit 3 carries out the desired and specified movement. The movementcontroller can be implemented in the control unit 10 but can also be anindependent unit. The manipulated variables SG for the active drivecoils 4 are then determined from the driving forces and/or drivingtorques to be impressed. The active drive coils 4 involved in theimplementation of the movement are thus energized such that the drivingforces and/or driving torques required for the specified movement aregenerated.

The control unit 10 can also take into account any failed drive coils 4and determine the manipulated variables SG for the active drive coils 4so that a failed drive coil 4 in the region of the drive magnetarrangement 8 is compensated in order to carry out the desired andspecified movement. The electromagnetic field required for the movementof the transport unit 3 is thus generated with the active drive coils 4excluding the failed coil.

This enables a method for operating a linear motor 1 comprising a stator2, on which a plurality of drive coils 4 is arranged, and comprising atransport unit 3, which is moved along the stator 2 and on which aplurality of drive magnets 9 is arranged, wherein an electromagneticfield, which interacts with the drive magnets 9 of the transport unit 3for moving the transport unit 3, is generated by energizing active drivecoils 4 in the region of the drive magnets 9 of the transport unit 3. Acontrol unit 10 determines manipulated variables SG, preferably coilcurrents i_(c) or coil voltages v_(c), for the active drive coils 4. Theactive drive coils 4 are energized according to the determinedmanipulated variables SG in order to control the movement of thetransport unit 3. This can take place in specified time steps of thecontrol of the movement. The control unit 10 is aware of a failed drivecoil 4 or itself detects a failed drive coil 4 in the region of thedrive magnet arrangement 8. For moving the transport unit 3, the effectof the failed drive coil 4 is compensated by the remaining active drivecoils 4. For this purpose, the control unit 10 generates the manipulatedvariables SG for the remaining active drive coils 4 so that for movingthe transport unit 3 the failed drive coil 4 is compensated. This can beperformed with an optimization as described below and claimed in theclaims.

The control unit 10 and/or a movement controller can bemicroprocessor-based hardware on which control software is run. However,the control unit 10 and/or a movement controller can also be implementedas computer software that is installed and run on available computerhardware. However, the control unit 10 and/or a movement controller canalso be designed as an integrated circuit, such as anapplication-specific integrated circuit (ASIC) or field programmablegate array (FPGA), on which a microprocessor can also be implemented.The control unit 10 can however also be designed as an analog circuit,for example as an analog computer. Combinations of these are alsopossible.

In order to control the movement of the transport unit 3, themanipulated variable SG, coil current i_(c) or preferably the coilvoltage v_(c), are thus determined for each time step of the control foreach active drive coil 4, with which the active drive coils 4 areenergized in order to generate the magnetic fluxes required for themovement.

In order to determine the manipulated variable SG, such as the coilcurrent i_(c) or the coil voltage i_(c), of the active drive coils 4, aquality functional J is used, which is a function f of at least onemanipulated variable SG of the active drive coils 4 to be determined.i.e., J=f(SG), wherein SG is in each case a vector with the manipulatedvariables to be determined of the individual active drive coils 4involved. Specifically, the quality functional J is a sum of a numberk≥1 of quality terms JT_(k) weighted with weighting factors k_(k), i.e.,

${J({SG})} = {\sum\limits_{k}{k_{k} \cdot {{{JT}_{k}({SG})}.}}}$

At least one, preferably each, quality term JT_(k) is a function of amanipulated variable SG of the active drive coils 4. A quadraticfunction is preferably used since this has a global minimum and istherefore well-suited for optimization. The quality functional J andthus also a quality term JT_(k) can however also be dependent on othervariables of the linear motor 1, for example on a position of thetransport unit 3 relative to the stator 2 or of magnetic variables (suchas the magnetic flux) in the stator 2.

If control unit 10 is aware whether and which drive coil 4 has failed,this can be taken into account in the quality functional J in the vectorof the manipulated variables SG. This would also automaticallycompensate for a failure of a drive coil 4.

This quality functional J(SG) of the control is optimized (maximized orminimized), generally minimized, in order to determine the manipulatedvariables SG for the active drive coils 4 involved, which manipulatedvariables are generated in the relevant time step in order to energizethe drive coils 4. Optimization means that the manipulated variables SGthat minimize or maximize the quality functional J(SG) are sought. Thesemanipulated variables SG are also referred to as optimal manipulatedvariables SG_(opt).

The optimization can be performed on-line, i.e., in each time step ofthe control, or also off-line. In off-line optimization, the manipulatedvariables SG to be set during the performance of the movement scenariocan be calculated in advance for a specific specified movement scenarioof the transport unit 3 (for example as a time curve of the position orspeed of the transport unit over time or over distance travelled). Forthis purpose, the movement scenario can be discretized in certain timesteps and the optimization can be performed in each of these time steps.Interpolation between the time steps may also be performed. Themanipulated variables SG are thus present at any required time point ofthe movement of the transport unit 3, in particular in each time step ofthe control. Since the movement of a transport unit 3 in a transportsystem is usually planned in advance, the movement scenario is known. ifthe movement scenario is not known or only partially known, the actualmovement of the transport unit 3 can be delayed so that sufficientinformation about the movement will be known again.

In the case of off-line optimization, movement characteristic maps canalso be generated. During the movement of the transport unit 3 along thestator 2 at a specific speed, which corresponds to a specific drivingforce acting on the transport unit 3 in the movement direction, thedrive magnets 8 of the transport unit 3 are moved past the drive coils4. Thus, the coil voltages v_(c) or coil currents i_(c) to be impressedinto the active drive coils 4 are repeated. This makes it possible tocreate off-line a movement characteristic map for the coil voltagesv_(c) or coil currents i_(c) for a specific movement scenario in theform of a combination of position and driving force. During the movementof the transport unit 3, the required coil voltages v_(c) and/or coilcurrents i_(c) must then be read out in the movement characteristic maponly as a function of the currently required driving force and thecurrent position. Such movement characteristic maps can also be createdfor different quality functionals J(SG).

Mathematically, the optimization can generally be written as

$\min\limits_{SG}{J({SG})}$

(wherein the cost function J as mentioned can also be dependent onvariables other than the manipulated variables SG).

The optimization can also take place taking into account a constraint g.The constraint g describes the physical laws of the linear motor 1.These laws are formulated in the form of a mathematical model of thelinear motor 1, preferably likewise as a function of the manipulatedvariable SG and optionally of further variables (such as statevariables) of the linear motor 1. The mathematical model can be a modelthat models the underlying physical laws of the linear motor 1, forexample the relationship between the coil current i_(c) or coil voltageu_(c) and the magnetic variables, such as the magnetic flux. However,the mathematical model can also be a trained model, for example a localmodel network or a neural network. In general, the model simulates theresponse of the linear motor 1 in the form of an output variable or astate variable (both usually vectors), or also both, to an inputvariable (usually a vector). The input variable is usually themanipulated variable SG, for example a coil current i_(c) or a coilvoltage u_(c). The output variable can, for example, be an actingdriving force (propulsive force and/or normal force) or driving torqueon the transport unit 3 but also a movement variable of the transportunit 3, such as a position along the stator 2, a speed in the movementdirection x and/or normal direction y or an acceleration in the movementdirection x and/or normal direction y. In the case of a planar motor.the driving forces can act in a movement plane, i.e., in an x-z plane.The output variable can also be a total current of the drive coils 4involved (such as a q- or d-current). According to kinematic principles,the driving forces/driving torques and the movement variables areequivalent, or in other words, the resulting movement variables are theconsequence of the acting forces/torques. The driving forces/drivingtorques correspond to a total current. The state variables, whichdescribe the state of the long-stator linear motor 1, are, for example,the magnetic variables resulting from the acting input variable in thelong-stator linear motor 1, such as the magnetic fluxes, magneticpotentials or voltages.

A failed drive coil 4 can also be formulated as an additional constraintg, for example in the form of a mathematical formulation for “drive coilwith the number xy has failed.” For example, this could be realized by avector for all drive coils 4, said vector containing a binary value foreach drive coil 4, wherein “0” means “failed,” for example, and “1”means “not failed.”

Mathematically, taking into account the constraint g, the optimizationcan generally be written as follows (wherein the cost function J and/orthe constraint g, as mentioned, may also be dependent on variables otherthan the manipulated variables SG).

$\min\limits_{SG}{J({SG})}$ g(SG) = 0

To solve such optimization problems, there is a plethora of knownsolution algorithms, such as the gradient method, Newton's method,evolutionary methods or sequential quadratic programming, to name but afew. However, the selection of the solution algorithm plays no role inthe invention, but a method that is advantageous with regard tocomputing effort and computing time (in particular in the case of anon-line optimization) is naturally selected. What the solution methodshave in common is that possible solutions of the optimization problemare sought, usually iteratively, until a defined termination criterionis achieved. The termination criterion can, for example, be a number ofiterations or the undershooting of a limit value of the differencebetween the solutions of two successive iteration steps of theoptimization problem, or also another termination criterion. Theselection of the solutions (i.e., the manipulated variables SG) in eachiteration step takes place by means of the rules of the solution method,wherein a suitable selection of the solution can be specified as thestarting value in the first iteration step. In the gradient method, forexample, the gradient of the quality functional (derivative of thequality functional with respect to the manipulated variable) isdetermined, and the manipulated variable for the next iteration step isselected along this gradient, wherein the step size from the currentmanipulated variable to the next manipulated variable is determined bythe specified rules of the solution method.

Since the movement of the transport unit 3 along the stator 2 is to becontrolled, the quality functional J(SG) contains a quality termJTk(SG), which evaluates, as explained in detail below, a deviation of aspecified movement setpoint variable BS of the control of the movementof the transport unit 3 from an actual variable IS of the control,preferably a movement actual variable, which is dependent on themanipulated variable SG. As usual in a control, the relationship is suchthat the manipulated variable SG is to be generated for each time stepof the control such that the actual variable IS, which is brought aboutby the manipulated variable SG, is to correspond as well as possible toa specified movement setpoint variable BS, usually expressed by aspecific maximum control error. The actual variable IS is usuallymeasured or determined from other measured variables.

In the case of linear motors 1 of a transport system, it is generallysuch that a transport unit 3 can be moved along the stator 2 over acertain route, also over several routes connected to one another viaswitches. A route is planned in advance, or even during movement, foreach transport unit 3, which can also comprise the planning of amovement profile (movement variable, such as speed over time or distancetravelled). Processing stations, in which an object being moved with thetransport unit 3 is processed, may also be provided along the route. Atransport unit 3 may also convey different or changing objects duringthe movement along the route, for example bottles that are filled. Inshort, the requirements on the movement of the transport unit 3 duringmovement along a route of the linear motor 1 can change.

In order to be able to map this in the control of the movement of thetransport unit 3 in a simple manner, it is provided that there are atleast two movement phases during the movement of the transport unit 3along the route, wherein different quality functionals J(SG) are used inthe movement phases. The quality functionals J(SG) can differ by thenumber k of quality terms JT_(k)(SG) used and/or by the quality termsJT_(k)(SG) used and/or by the weighting factors k_(k). The control ofthe movement of the transport unit 3 in the various movement phases canthus be adapted in a situation-related manner and flexibly to differentrequirements. It is even possible to do so during the operation of thelinear motor 1.

For example, in a first movement phase, in which no particular accuracyis required in the control of the movement of the transport unit 3,energy-optimal operation of the long-stator linear motor 1 can besought, while in a second movement phase, in which an accurate controlof the movement is required, an accurate control of the driving forcesmay be desired. For example, in the region of a switch, the control ofthe movement of the transport unit 3 can take place with a higherweighting of the control of the normal forces (in the y-directiontransverse to the movement direction x). Outside the switch, however,with higher energy efficiency.

A quality term JT_(k)(SG) evaluates the deviation of the setpointvariable BS of the control from the actual variable IS of the control,preferably as a square of the deviation, i.e., for example,JT_(k)(SG)=k_(e)(BS−IS(SG))² or JT_(k)(SG)=k_(e)(BS−IS(SG)), with theweighting factor k_(e). Such a quality term JT_(k)(SG) can be used inparticular in an on-line optimization where the current actual variablesIS are available (for example by measurement). The actual variable IScan be measured, for example with position sensors 13, but can also bedetermined from existing known measurement variables, for example withan observer, wherein the mathematical model of the linear motor 1 canalso be the basis for the observer. In the case of an off-lineoptimization, the actual variable IS can be determined from themathematical model (i.e., output variable or state variable of themodel) of the linear motor 1. Since the actual variable IS is dependenton the manipulated variable SG, this quality term of JT_(k)(SG) is alsoindirectly dependent on the manipulated variable SG. With this, forexample, the deviation of a setpoint speed from an actual speed or thedeviation of a setpoint position from an actual position or thedeviation of a setpoint driving force (or also torque) from an actualdriving force (or also driving torque) or the deviation of a setpointmagnetic flux from an actual flux can be evaluated. This quality termJT_(k)(SG) accordingly evaluates the control error. Since the positionand the speed are naturally a consequence of the driving force anddriving torque acting on the transport unit 3, and the driving force anddriving torque are dependent on the magnetic flux, the deviationsmentioned are equivalent.

The driving force acting on the transport unit 3 or the acting drivingtorque can be obtained from the mathematical model of the constraintg(SG), wherein the driving force can be a propulsive force (also in anx-z movement plane) and/or a normal force but also a force in thez-direction. In general, the driving torque can be a torque about one ofthe axes x, y, z. The driving force and the driving torque can be avector with components of the different axes x, y, z. The actual drivingforce or the actual driving torque can then be calculated from themodel. The setpoint driving force or the setpoint driving torque isdetermined, for example, from the movement setpoint variable BS of thecontrol or can also be specified directly as a setpoint variable.Preferably, a movement controller is implemented, which determines therequired setpoint driving force or the required setpoint driving torquefrom the movement setpoint variable BS and a current movement actualvariable IS (such as an actual position or actual speed) on the basis ofan implemented control rule (e.g., a PI or PID controller).

Another quality term JT_(k)(SG) could, for example, evaluate theelectrical power required for the movement of the transport unit 3.Since the coil currents i_(c) or coil voltages v_(c) can be used as themanipulated variable SG, the electrical power can be easily determinedfrom the manipulated variables SG. Since the coil currents i_(c) or coilvoltages v_(c) can be used as the manipulated variable SG, a suitablequality term JT_(k)(SG) for evaluating the power is the square of theEuclidean norm ∥·∥₂ ² of the manipulated variables SG (which can becombined into a vector) of the involved energized active drive coils 4,i.e., for example, JT_(k)(SG)=k_(i)·∥SG∥₂ ², with the weighting factork_(i). The Euclidean norm is known to be the root of the sum of thesquares of the components of the vector of the manipulated variables SG.The square thereof is consequently the sum of the squares of thecomponents of the vector. Since the electrical power is proportional tothe square of the electrical current or of the electrical voltage, thesquare of the Euclidean norm ∥·∥₂ ² evaluates the implemented electricalpower, and thus also the energy efficiency.

Another quality term JT_(k)(SG) could be the sum SG_(Σ) of the values ofthe manipulated variables SG of the drive coils 4 involved in themovement of the transport unit 3, also as a square of the sum, i.e., forexample, JT_(k)(SG)=k_(Σ)SG_(Σ) ² or JT_(k)(SG)=k_(Σ)SG_(Σ), with theweighting factor k_(Σ). It can thus be achieved that the sum of themanipulated variables is evaluated. The smaller the sum of themanipulated variables SG, the lower the load on the power electronicssystem for generating the manipulated variables SG. This can beadvantageous for the long-term operation of the long-stator linear motor1 because the power electronics system is loaded as uniformly aspossible. In particular, the thermal load on the power electronicssystem or the circuit components in the power electronics system canthus also be reduced because of the smaller losses. It can thus also beachieved that the stator 2 is heated less, which can also simplify oreliminate cooling of the stator 2.

A possible quality functional J(SG) could thus be defined as follows.

J(SG)=(k _(i) ∥SG∥ ₂ ² +k _(Σ) SG _(Σ) ² +k _(e)(BS−IS(SG))²)

If the propulsive force F_(x) and the normal force F_(z) are controlled,the quality functional J(SG) can, for example, be written as follows.

J(SG)=(k ₁ ∥SG∥ ₂ ² +k _(Σ) SG _(Σ) ² +k _(x)(F _(xS) −F _(x)(SG))² +k_(y)(F _(yS) −F _(y)(SG))²)

In the case of a planar motor, a driving force F_(z) can also act in thez-direction and a corresponding quality term JT_(k) can also be providedin the quality functional J(SG). Therein, F_(xS), F_(yS), F_(zS) are thespecified setpoint values of the forces and F_(x)(SG), F_(y)(SG),F_(z)(SG) are the actual values of the forces determined using themathematical model. Naturally, the quality functional J(SG) can alsoinclude only the propulsive force F_(x), F_(z) or only the normal forceF_(y). In the same way, additional or other components of the drivingforce or also of the driving torque can naturally also be taken intoaccount.

It is obvious that by changing the weighting factors k_(k), for examplek_(i), k_(Σ), k_(e), k_(x), k_(y), k_(z), and/or by adding or removingspecific quality terms JT_(k)(SG), for example ∥SG∥₂ ², SG_(Σ) ²,(F_(xS)−F_(x)(SG))², (F_(yS)−F_(y)(SG))², the control objective and thebehavior of the control can be changed in a simple manner. Naturally,quality terms JT_(k)(SG) other than or in addition to the ones describedabove are also possible.

By (on-line or off-line) optimization of the quality functional J(SG)with respect to the manipulated variables SG_(opt), optimal manipulatedvariables SG_(opt) can consequently be determined, which minimize (ormaximize) the quality functional J(SG). These optimal manipulatedvariables SG_(opt) are implemented on the linear motor 1 in each timestep of the control of the movement of the transport unit 3, asschematically illustrated in FIG. 6 . The movement setpoint variable BSis, for example, a specification for a desired propulsive force F_(x),F_(z) and normal force F_(y), or a position of the transport unit 3 tobe approached or a speed of the transport unit 3 to be set. Compliancewith this movement setpoint variable BS is, for example, ensured by thequality terms (F_(xS)−F_(x)(SG))², (F_(yS)−F_(y)(SG))²,(F_(zS)−F_(z)(SG))², (BS−IS(SG))², or other or additional components ofthe driving force or the driving torque, wherein the accuracy ofcompliance can be influenced by means of the associated weightingfactors k_(x), k_(y), k_(z), k_(e). The other quality terms would nothave to be present at all or could be weighted lower in comparison. Ifhigh accuracy is not required, the quality term ∥SG∥₂ ², whichcharacterizes the electrical power, in particular power loss, could beweighted higher or added in order to enable the most energy-savingoperation possible. The quality functional J(SG) is thus changed indifferent movement phases during the movement of the transport unit 3.The different movement phases with the different quality functionalsJ(SG) can be configured in advance or can also first be defined duringthe movement, for example by the system control unit 11.

By providing further quality terms JT_(k), other aspects of theoperation of the linear motor 1 can naturally also be taken intoaccount.

With reference to FIG. 6 and FIG. 7 , possible exemplary embodiments forthe implementation of the determined optimal manipulated variablesSG_(opt) on the linear motor 1 or on the active drive coils 4 of thelinear motor 1 are described.

In the example according to FIG. 7 , the optimal manipulated variableSG_(opt), for example coil voltages v_(c,opt) of the active drive coils4, determined by the optimization is directly generated by the powerelectronics system (not shown) and applied to the drive coils 4. In thisexample, the movement setpoint variable BS is a setpoint positionx_(set) of the transport unit 3. This setpoint position x_(set) can alsobe used in the optimization, for example for a cost termJT_(k)(SG)=k_(e)(BS−IS(SG))². For this purpose, an actual value IS ofthe movement variable can also be determined (for example measured orcalculated), for example a current actual position x in this case, andcan be used in the optimization. From the movement setpoint variable BSand the actual value IS of the movement variable, a driving force F_(xS)is determined in a movement controller RB, in this example a positioncontroller, in order to minimize the control error between actual valueIS and movement setpoint variable BS in the current time step of thecontrol. This driving force F_(xS) can, for example, be used for a costter (F_(xS)−F_(x)(SG))² in the optimization. The setpoint value F_(xS)of the driving force could, however, also be specified as the movementsetpoint variable BS. In this case, the movement controller RB couldalso be implemented separately from the control unit 10, for example asmicroprocessor-based hardware with software running thereon or as anintegrated or analog circuit. In the optimization, in particular in themathematical model of the linear motor 1 for the constraint g(SG) of theoptimization, an actual value of the coil currents i_(c) of the activedrive coils 4, which can be measured or determined from other knownvariables, can also be used. This embodiment of the control unit 10 issuitable in particular for on-line optimization because current actualvalues are captured on the linear motor 1 in each time step of thecontrol, for example an actual position x and/or an actual value of thecoil current i_(c) (or also the coil voltage v_(c)). However, thisembodiment can also be used for an off-line optimization.

In the exemplary embodiment according to FIG. 8 , the determined optimalmanipulated variables SG_(opt) are not directly generated andimplemented but are adjusted on the linear motor 1. The optimalmanipulated variable SG_(opt) determined with the optimization are, forexample, the optimal coil currents i_(c,opt) of the active drive coils4. These optimal coil currents i_(c,opt) are used in the control unit 10as control variables in a manipulated variable controller RS. From theoptimal manipulated variables SG_(opt), here the coil currentsi_(c,opt), and current actual values of the manipulated variable SG,here the coil currents i_(c) of the active drive coils 4, specificallyfrom the deviations thereof, the manipulated variable controller RSdetermines, on the basis of an implemented control rule (e.g., a PI orPID controller), the manipulated variable SG, here coil voltages v_(c)of the active drive coils 4, in each time step of the control. Thesemanipulated variables SG, here coil voltages v_(c), are generated andapplied to the drive coils 4 by a power electronics system (not shown).Naturally, such a manipulated variable controller RS exists for each ofthe active drive coils 4. The manipulated variable controller RS can bedesigned as software, microprocessor-based hardware or as an integratedor analog circuit. However, the manipulated variable controller RS couldalso be part of the power electronics system. In addition, optimal coilvoltages v_(c,opt) could also be determined as optimal manipulatedvariables SG_(opt) in the optimization. These optimal coil voltagesv_(c,opt) could then be used as feedforward control of the manipulatedvariable controller RS, as indicated by dashed lines in FIG. 8 . Theoptimal coil voltages v_(c,opt) and the coil voltages determined withthe manipulated variable controller RS are added up to the manipulatedvariable SG, the coil voltages v_(c) to be applied. In this exemplaryembodiment too, a movement controller RB could be provided analogouslyto the embodiment according to FIG. 7 in order to determine the drivingforce F_(xS) from a movement setpoint variable BS, for example asetpoint position.

The embodiment of the control unit 10 according to FIG. 8 is inparticular suitable for off-line optimization but could also be used foron-line optimization.

In the case of off-line optimization, movement characteristic maps 12created in advance could be used, which map the driving force F_(x)(and/or F_(y), F_(z)) and a position x of the transport unit 3 relativeto the drive coils 4 onto optimal manipulated variables SG_(opt), forexample coil currents i_(c,opt) and/or coil voltages v_(c,opt), asalready explained above. FIG. 9 shows, by way of example, such amovement characteristic map 12, which maps the position x of thetransport unit 3 relative to a plurality of active drive coils 4, and adriving force F_(x) to the coil currents i_(c) of the active drive coils4. In this case, movement characteristic maps 12 for differently definedquality functionals J(SG) can naturally be used in order to determinethe optimal manipulated variables SG_(opt) in different movement phases.The movement characteristic maps 12 can be stored in the control unit 10or in external memory units. For the control of the movement of thetransport unit 3, the optimal manipulated variables SG_(opt) must thenonly be read out in the correct movement characteristic map. Very littlecomputing capacity is required for this purpose.

The movement characteristic maps 12 can be stored in the form of tables.Interpolation between the table entries could then be performed.However, it is also possible to adapt a mathematical function, forexample a polynomial of a specific order, to the determined entries in amovement characteristic map 12, for example by means of a curve-fittingmethod. In this case, only the specific mathematical function would haveto be stored for the movement characteristic map 12 and interpolationwould no longer be necessary.

Equivalent controls can naturally be realized for other movementdirections, for example in the y- or z-direction.

A variable of a quality term JT_(k)(SG), for example F_(x)(SG),F_(y)(SG) or F_(z)(SG) in (F_(xS)−F_(x)(SG))², (F_(yS)−F_(y)(SG))²,(F_(zS)−F_(z)(SG))², can be determined using a mathematical model of thelinear motor 1. In this case, the manipulated variable SG serves as aninput variable of the model and the variable is an output variable orstate variable of the model.

The mathematical modeling of the linear motor 1 for the constraint g(SG)of the optimization and/or for the determination of a variable of aquality term JT_(k)(SG) on the basis of given physical conditions can becarried out in different ways. A common and sufficiently known model oflinear motor 1 is a dq-model or a finite element (FEM) model.

A dq-model is, for example, described in the document cited at thebeginning by Khong, P. C., et al. but also in Deng, Z., et al., “Forcesand parameters of permanent magnet linear synchronous machines,” IEEETransactions on Magnetics, Vol. MAG-23, No. 1, January 1987 or inChapter 4 of Boldea, I., et al., “Linear Electric Actuators andGenerators,” Cambridge University Press, 1997. The dq-model describesthe relationship between the currents, voltages and magnetic fluxes inthe dq-coordinate system and the manipulated variables SG (coil currentsi_(c) and coil voltages v_(c)) and also comprises equations for theacting forces in the movement direction x (driving force) and in thetransverse direction y (normal force), and also of other force or torquecomponents.

Another advantageous possibility for modeling the behavior of the linearmotor 1 is by means of a reluctance network with which the relationshipbetween coil currents i_(c) or coil voltages v_(c) and the magneticfluxes, and also the generated driving forces, propulsive force and/ornormal force, and driving torques can be described. The advantage ofmodeling by means of a reluctance network is that all forms of the motorvariables are taken into account and not just the sinusoidal fundamentalwaves. As a result, non-linear effects, such as saturation, but alsoeffects such as a cogging force can be modeled systematically butnevertheless with sufficient abstraction so that the model can also becalculated on-line.

The use of a reluctance network model of the linear motor 1 for thecontrol of the movement of a transport unit 3, also with other controlmethods as described above, is in its own right novel and inventive. Forexample, the reluctance network model could be used to create anobserver to estimate actual values of the magnetic fluxes and/or of theforces which are then used in a control.

A reluctance network describes the linear motor 1 as a network ofreluctances R (magnetic resistances), permeances G (magnetic conductancevalues) and magnetic voltage sources. A reluctance as is known describesthe relationship between a magnetic voltage and a magnetic flux Φ. Thepermeance G is the reciprocal of the reluctance R. The reluctance R of amagnetic conductor of length I and cross-section A is given by

$R = {\frac{\text{?}}{\text{?}\mu_{0}A}.}$?indicates text missing or illegible when filed

where μ₀ is the permeability of vacuum and μ_(r) is the relativepermeability of the material of the magnetic conductor.

Modeling with a reluctance network is described using the example of along-stator linear motor as a linear motor 1 on the basis of a structureof the long-stator linear motor as shown in FIG. 2 . In the case ofother structural designs or other linear motors 1, such as in the caseof a planar motor, corresponding changes can result in the reluctancenetwork and in the model.

The stator 2 is modeled as a reluctance network RN_(s) with i=1, . . . nteeth (stator teeth 5 and secondary teeth 7). These teeth do not have tobe all of the teeth present on the stator 2, but only the teethinteracting with the transport unit 3, for example. Usually, a number ofdrive coils 4 are selected, which are energized for the purpose ofmoving the transport unit 3. This gives the number n of teeth, which aremodeled with the reluctance network RN_(s). On the j=1, . . . , n_(c)stator teeth 5, where n_(c)≤n, drive coils 4 are arranged, between whichthe secondary teeth 7 are arranged. In the case of no secondary teeth 7,the reluctance network RN_(s) is simplified accordingly. The useddimensions of the stator 2 and of the transport unit 3 are specified inFIG. 2 and can be assumed to be known for a linear motor 1.

Each stator tooth 5 is modeled by a differential reluctance (reluctanceper unit length)

${\text{?}( \Phi_{i} )} = {\frac{\text{?}}{\mu_{0}{\mu_{r}( \Phi_{i} )}A_{c}}.}$?indicates text missing or illegible when filed

with the cross-sectional area A_(c)=w_(c)·b_(s) of the stator tooth 5.By means of the non-linear modeling of the material by the relativepermeability μ_(r) dependent on the magnetic flux Φ, the magneticsaturation of the stator tooth 5 can be taken into account. Analogously,each secondary tooth 7 is modeled by a differential reluctance

${\text{?}( \text{?} )} = {\frac{\text{?}}{\mu_{0}\text{?}( \Phi_{i} )\text{?}}\text{?}}$?indicates text missing or illegible when filed

with the cross-sectional area A_(a)=w_(a)·b_(s) of the secondary tooth5. The coupling of the teeth through the tooth interspace of widthw_(ca) between two adjacent teeth is modeled by a differential permeance

$G^{\prime} = {\frac{\mu_{0}\text{?}}{\text{?}}.}$?indicates text missing or illegible when filed

The j drive coils 4 can be modeled as differential magnetic voltages

$\text{?} = {\frac{N_{c}}{\text{?}}{i_{c,j}.}}$?indicates text missing or illegible when filed

where N_(c) denotes the number of windings of the drive coil 4 and I_(s)denotes the length of the tooth in the y-direction (i.e., transverselyto the movement direction x).

A differential section dy of the stator 2 in the y-direction can then bemodeled with a reluctance network RN_(s) of the stator 2 as shown inFIG. 3 . Each tooth therein forms a branch of the reluctance network ofthe stator RN_(s), wherein the branches are connected by thedifferential permeances G′ so that nodes of the reluctance networkRN_(s) form. Therein, Φ_(i)(y) denotes the magnetic flux in the i-thtooth (stator tooth 5 or secondary tooth 7) and φ_(i)(y) denotes themagnetic potential in the y-direction along the i-th tooth.

Thus, at each of the n nodes of the reluctance network of the statorRN_(s), a node sum of the magnetic fluxes Φ_(i)(y) and of the magneticpotential φ_(i)(y) along the tooth in the y-direction can be formulated,leading to the following 2n equation systems as system equations.

Φ₁(y)−Φ₁(y+dy)−G′dy(φ₁(y+dy)−φ₂(y+dy))=0

Φ₂(y)−Φ₂(y+dy)+G′dy(φ_(i)(y+dy)−φ₂(y+dy))−G′dy(φ₂(y+dy)−φ₃(y+dy))=0

Φ_(a)(y)−Φ_(n)(y+dy)−G′dy(φ_(n−1)(y+dy)−φ_(n)(y+dy))=0

φ₁(y)−φ₁(y+dy)−R′ _(a) dyΦ ₁(y)=0

φ₂(y)−φ₂(y+dy)−R′ _(c) dyΦ ₂(y)+u′_(c,i) dy=0

φ_(n)(y)−φ_(n)(y+dy)−R′ _(a) dyΦ _(n)(y)=0

The local, differential description of the flux or potential variablesmakes it possible, in particular, to take into account non-negligibleleakage fluxes in the tooth interspace.

Dividing the equations by dy and taking into account dy→0 yields thesystem equations

${\frac{d}{dy}\begin{bmatrix}{\Phi(y)} \\{\varphi(y)}\end{bmatrix}} = {{\begin{bmatrix}0 & A_{1} \\{A_{2}(\Phi)} & 0\end{bmatrix}\begin{bmatrix}{\Phi(y)} \\{\varphi(y)}\end{bmatrix}} + {\begin{bmatrix}0 \\B\end{bmatrix}{i_{c}.}}}$

The n magnetic fluxes and magnetic potentials are combined in thevectors Φ(y) and φ(y), i.e., Φ=[Φ₁ Φ₂ . . . Φ_(n)]^(γ) and φ=[φ₁ φ₂ . .. φ_(n)]^(γ). The system matrices A₁ and A₂ result as

$A_{1} = \begin{bmatrix}{- G^{\prime}} & G^{\prime} & 0 & \cdots & & & 0 \\G^{\prime} & {{- 2}G^{\prime}} & G^{\prime} & \cdots & & & 0 \\0 & G^{\prime} & {{- 2}G^{\prime}} & G^{\prime} & 0 & \cdots & 0 \\ \vdots & & & \ddots & & & \vdots \\0 & \cdots & & \cdots & 0 & G^{\prime} & {- G^{\prime}}\end{bmatrix}$

and A₂(Φ)=diag[−R′_(a)(Φ₁),−R′_(c)(Φ₂), . . . , −R′_(a)(Φ_(n))] (whereindiag denotes a diagonal matrix) and the input matrix B results as

$B = {{\frac{N_{c}}{\text{?}}\begin{bmatrix}0 & 0 & \cdots & 0 \\1 & 0 & \cdots & 0 \\0 & 0 & \cdots & 0 \\0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & 0\end{bmatrix}}.}$ ?indicates text missing or illegible when filed

In order to fully model the stator 2, the terminations at y=I_(s) (i.e.,the stator yoke 6) and at y =0 (i.e., at the transition to the air gap)must still be modeled as a boundary condition, as explained withreference to FIG. 4 . The stator yoke 6 at y=I_(s) between two adjacentteeth is modeled with a linear permeance G_(sy), with

$\text{?} = {\mu_{0}\text{?}{\frac{2\text{?}}{w_{a} + w_{c} + {2w_{ca}}}.}}$?indicates text missing or illegible when filed

where μ_(ry) is the relative permeability of the stator yoke 6 assumedto be constant and A_(sy)=b_(s)·I_(j) is the cross-sectional area of thestator yoke 6. At y=0, the teeth are coupled by leakage permeancesG_(s0), which can, for example, be modeled with

$G_{s0} = {\frac{\mu_{0}\text{?}}{\pi}{{\ln( \frac{2\text{?}}{w_{ca}} )}.}}$?indicates text missing or illegible when filed

The linear permeances G_(sy) and the leakage permeances G_(s0) againconnect the branches of the reluctance network of the stator RN_(s) toform further nodes in the reluctance network of the stator RN_(s). Themagnetic flux Φ_(Σ)=[Φ_(Σ1) Φ_(Σ2) . . . Φ_(Σn)]^(T) denotes themagnetic flux of the drive magnets 9 of the transport unit 3 that flowsinto the teeth of the stator 2, i.e., the magnetic flux in the air gap.

With the same procedure as described above, equation systems

Φ(I _(s))+G _(sy) T _(L) ^(φ)φ(I _(y))=0

Φ(0)+Φ_(Σ) +G _(s0) T _(L) ^(φ)φ(0)=0

with the matrix

$\text{?} = \begin{bmatrix}{- 1} & 1 & 0 & \cdots & & & 0 \\1 & {- 2} & 1 & \cdots & & & 0 \\0 & 1 & {- 2} & 1 & 0 & \cdots & 0 \\ \vdots & & & \ddots & & & \vdots \\0 & \cdots & & \cdots & 0 & 1 & {- 1}\end{bmatrix}$ ?indicates text missing or illegible when filed

are again obtained as boundary conditions for y=I_(s) and y=0 andsupplement the system equations.

If the material of the teeth of the stator 2 is assumed to bemagnetically linear, the dependence of the reluctances R′_(a) and R′_(c)on the magnetic flux Φ drops out and an analytical solution for theabove differential system equations can be given.

The linked fluxes Ψ_(c)=[Ψ_(c1) Ψ_(c2) . . . Ψ_(cn) ₀ ]^(T) in the j=1,. . . , n_(c) stator teeth 5 of the stator 2 then result as

$\text{?} = {\frac{N_{c}}{\text{?}}\text{?}{\Phi(y)}{{dy}.}}$?indicates text missing or illegible when filed

In the integral, the matrix T_(c) ^(Ψ)=[e₂ e₄ . . . e_(n−1)]^(T) withthe i-th unit vectors e_(i) selects the magnetic fluxes associated witheach stator tooth 5 in Φ (y).

If, on the other hand, the material is non-linear and magneticsaturation occurs. an analytical solution is not possible. In this case,approximation methods for solving the differential equations of thesystem equations can be used.

For example, the i-th magnetic flux Φ_(i) and the i-th magneticpotential φ_(i) in the teeth can be approximated by means ofapproximation polynomials g_(j)(y) with the coefficients {circumflexover (Φ)}_(ij) and {circumflex over (φ)}_(ij) via a number N ofinterpolation nodes along the length I_(s) (in the y-direction) of theteeth, in the form

${{\Phi_{i}(y)} \approx {\sum\limits_{j = 0}^{N}{{\hat{\Phi}}_{ij}{g_{j}(y)}}}} = {{g^{T}(y)}{\hat{\Phi}}_{i}}$${{\varphi_{i}(y)} \approx {\sum\limits_{j = 0}^{N}{{\hat{\varphi}}_{ij}{g_{j}(y)}}}} = {{g^{T}(y)}{{\hat{\varphi}}_{i}.}}$

By suitable selection of the approximation polynomials g_(j)(y), thesystem of non-linear differential equations in the system equations canbe transformed into a solvable system of non-linear algebraic equations.One possible selection of approximation polynomials g_(j)(y) areLagrange interpolation polynomials of the form

${g_{j}(y)} = {\text{?}\frac{y - y_{k}}{y_{j} - y_{k}}}$?indicates text missing or illegible when filed

where j=0, . . . , N. The transformation results by differentiation ofthe approximation polynomials g_(j)(y) with respect to y and evaluationat the N supporting points. A constant differentiation matrix D is thusobtained from

${\frac{d}{dy}{g^{T}(y)}\text{?}} = {{D\text{?}{or}\frac{d}{dy}{g^{T}(y)}{\hat{\varphi}}_{i}} = {D{{\hat{\varphi}}_{i}.}}}$?indicates text missing or illegible when filed

The matrix D thus has the entries

$d_{ij} = {( {\frac{d}{dy}{g_{j}(y)}} )_{y = y_{i}}.}$

The system of non-linear algebraic equations as system equations thenfollows as

${{\begin{bmatrix}\overset{\sim}{D} & 0 \\0 & \overset{\sim}{D}\end{bmatrix}\begin{bmatrix}\hat{\Phi} \\\hat{\varphi}\end{bmatrix}} - {\begin{bmatrix}0 & {\overset{\sim}{A}}_{1} \\{{\overset{\sim}{A}}_{2}( \overset{\sim}{\Phi} )} & 0\end{bmatrix}\begin{bmatrix}\hat{\Phi} \\\hat{\varphi}\end{bmatrix}} + {\begin{bmatrix}0 \\\overset{\sim}{B}\end{bmatrix}i_{c}}} = 0$

where {tilde over (D)}=diag[D D . . . D], {circumflex over(Φ)}=[{circumflex over (Φ)}₁ {circumflex over (Φ)}₂ . . . {circumflexover (Φ)}_(n)] and {circumflex over (φ)}=[{circumflex over (φ)}₁{circumflex over (φ)}₂ . . . {circumflex over (φ)}_(n)]. The transformedsystem matrices result as Ã₂({circumflex over (Φ)})=−diag[{circumflexover (R)}_(a)({circumflex over (Φ)}₁) {circumflex over(R)}_(c)({circumflex over (Φ)}₂) . . . {circumflex over(R)}_(a)({circumflex over (Φ)}_(n))] with {tilde over(R)}_(a)({circumflex over (Φ)}_(i))=−diag[R′_(a)({circumflex over(Φ)}_(i1)) R′_(a)({circumflex over (Φ)}_(i2)) . . . R′_(a)({circumflexover (Φ)}_(iN))] and {tilde over (R)}_(c)({circumflex over(Φ)}_(i))=−diag[R′_(c)({circumflex over (Φ)}_(i1)) R′_(c)({circumflexover (Φ)}_(i2)) . . . R′_(c)({circumflex over (Φ)}_(iN))], as well asÃ_(i)=A_(L)⊗I with the unit matrix I and {tilde over (B)}=B⊗I with I=[I1 . . . 1]^(T) and the Kronecker product ⊗.

The equations for the boundaries can also be transformed in this way,which results in the transformed equations

T _(L{circumflex over (Φ)}+) G _(sy) T _(L) ^(φ) T _(L){circumflex over(φ)}=0

−T ₀{circumflex over (Φ)}+Φ_(Σ) +G _(s0) T _(L) ^(φ) T ₀{circumflex over(φ)}=0

with the selection matrices T_(L) and T₀, which select the entries of{circumflex over (Φ)} and {circumflex over (φ)} at the boundaryinterpolation points (y=0 and I_(s)).

Thus, an approximation of the linked fluxes Ψ_(c)=[Ψ_(c1) Ψ_(c2) . . .Ψ_(cn)]^(T) can also be expressed as a function of the approximatedmagnetic fluxes {circumflex over (Φ)}, which can generally be specifiedas Ψ_(c)=W{circumflex over (Φ)}, wherein the matrix W results from theselection of the approximation polynomials g_(j)(y) and the Ninterpolation points.

In order to determine the magnetic flux generated by the drive magnets 9of the transport unit 3 in the air gap, the air gap and the transportunit 3 are also modeled with a reluctance network of the rotor RN_(L) asexplained with reference to FIG. 5 .

The transport unit 3 is coupled to the stator 2 by air gap permeancesG_(a,ij), with the subscript i=1, . . . , n for the number of teeth andthe index j=1, . . . , p for the number of drive magnets 9 on thetransport unit 3. The stator 2 is represented by the magnetic voltagesu_(s)=[u_(s,1) u_(s,2) . . . u_(s,n−1)]^(T), which result from themagnetic potentials φ at the edge of the stator 2 at y=0. With theapproximation explained above, the magnetic voltages result, forexample, from u_(s)=T_(s) ^(φ)T_(b){circumflex over (φ)} with the matrix

$\text{?} = \begin{bmatrix}1 & {- 1} & 0 & \cdots & & 0 \\0 & 1 & {- 1} & 0 & & 0 \\ \vdots & & \ddots & \ddots & & \vdots \\0 & \vdots & & 0 & 1 & {- 1}\end{bmatrix}$ ?indicates text missing or illegible when filed

and the selection matrix T₀ explained above.

The drive magnets 9 are, for example, permanent magnets, which aredescribed by voltage sources with constant magnetic voltagesu_(msj)=H_(c)I_(m), with the thickness I_(m) of the permanent magnetsand the magnetic coercivity H_(c) known for the permanent magnets used,and a linear permeance

$\text{?} = {\frac{A_{m}\mu_{0}\text{?}}{\text{?}}.}$?indicates text missing or illegible when filed

where A_(m) is the known cross-sectional area of the permanent magnetsand μ_(rm) is the known constant relative permeability of the permanentmagnets. Another model, even with non-constant magnetic voltages, can beused for other drive magnets 9. The magnet yoke connecting the drivemagnets 9 to the transport unit 3 is described with a linear permeance

$G_{b} = {\frac{\text{?}\mu_{0}\text{?}}{\text{?}}.}$?indicates text missing or illegible when filed

with the known cross-sectional area A_(t) and the known constantrelative permeability μ_(rt) and the distance w_(t) between two drivemagnets 9 (for example center-to-center as in FIG. 2 ). The leakage fluxat the end of the transport unit 3 facing the stator 2 is described byleakage permeances

$\text{?} = {\frac{\mu_{0}\text{?}}{\pi}{{\ln( \frac{2l_{g}}{\text{?}} )}.}}$?indicates text missing or illegible when filed

with the width w_(ms) of the interspace between two drive magnets 9.Each drive magnet 9 accordingly forms branches of the reluctance networkof the rotor RN_(L) with the linear permeance G_(m) and with themagnetic voltage u_(msj) of a magnetic voltage source, which areconnected at the ends of the branches by the leakage permeances G_(ml)at the air gap and the linear permeances at the magnet yoke to formnodes of the reluctance network of the rotor RN_(L).

The nodes at the air gap are linked to further branches of thereluctance network of the rotor RN_(L), in which air gap permeancesG_(a,ij) are arranged. Each of the p drive magnets 9 is connected via arespective air gap permeance G_(a,ij) to each of the n teeth taken intoaccount in the reluctance network of the stator RN_(s). The brancheswith the air gap permeances G_(a,ij) are connected by voltage sources tothe magnetic voltages u_(s), which represent the stator 2, torespectively form further nodes of the reluctance network of the rotorRN_(L).

The air gap permeances G_(a,ij)(x_(s), y_(s)), which describe themagnetic coupling between the stator 2 and the transport unit 3, aredependent on the position x_(s), y_(s) of the transport unit 3 in themovement direction x and in the transverse direction y relative to thestator 2. Each drive magnet 9 is magnetically coupled to each consideredtooth of the stator 2, or in other words, each drive magnet 9contributes to the magnetic flux Φ_(Σ) flowing into a tooth. The air gappermeances G_(a,ij)(x_(s), y_(s)) can be assumed to be known, forexample in that they are determined in advance by means of measurementtechnology or from simulation calculations. For this purpose, it is onlynecessary to determine the air gap permeances G_(a,ij)(x_(s), y_(s)) ofa drive magnet 9 as a function of the position x_(s), y_(s) relative toa stator tooth 5. If secondary teeth 7 are used, then also relative to asecondary tooth 7. The air gap permeances G_(a,ij)(x_(s), y_(s)) canthen be stored in the control unit 10, for example in tabular form, as amathematical formula or as characteristic maps.

It would also be possible to map the magnetic coupling between twoadjacent transport units 3 in the reluctance network of the rotorRN_(L). Such a dependence would thus also be taken into account by thecontrol.

In order to derive an equation system of independent equations from themagnetic reluctance network RN_(L) of the air gap and the transport unit3, different procedures can be used. The use of the well-known graphtheory is suitable as a possible approach, as explained by way ofexample below, The use of graph theory is advantageous because it offersa system for formulating and solving the equations with respect to theindependent variables.

The topology of the reluctance network of the rotor RN_(L) (FIG. 5 ) issubdivided into a tree and a co-tree. In this case, the network of thetree connects all nodes without forming a mesh (a circuit closed viabranches), and the co-tree comprises all elements that are not part ofthe tree. The selection of the tree is substantially free, but allmagnetic voltage sources are preferably located in the tree. A suitablechoice is, for example, the combination of all air gap permeancesG_(a,ij)(x_(s), y_(s)) except for the last air gap permeance G_(a,np)and the leakage permeances G_(ml) in the co-tree. The magnetic fluxesΦ_(t) of the tree and Φ_(c) of the co-tree can then be combined in thevectors Φ_(i) ^(T)=[Φ_(ts) ^(T) Φ_(tm) ^(T) Φ_(tg) ^(T)] and Φ_(c)^(T)=[Φ_(a,31) . . . Φ_(a,(n−1)p) Φ_(ml1) . . . Φ_(ml(p−1))]. The treeelements can be divided into the magnetic voltage sources of the stator2 with subscript ts, the magnetic voltage sources of the drive magnets 9with subscript tm, and the permeances in the tree with subscript tg.Consequently, the sub-vectors of the magnetic fluxes Φ_(ts)^(T)=[Φ_(s,1) Φ_(s,2) . . . Φ_(s,n−1)], Φ_(tm) ^(T)=[Φ_(ms1) Φ_(ms2) . .. Φ_(msp)], Φ_(t) ^(T)=[Φ_(m1) . . . Φ_(mp) Φ_(b1) . . . Φ_(b(p−1))Φ_(a,mp)] result. In an analogous manner, the vector of the magneticvoltages u_(t) of the tree and of the magnetic voltages of the co-treeu_(c) results, with the vectors of the magnetic voltages u_(ts)^(T)=[u_(s,1), u_(s,2), . . . , u_(s,n−1)], u_(tm) ^(T)=[u_(ms1),u_(ms2), . . . , u_(msp)] and u_(tg) ^(T)=[u_(m1), . . . , u_(mp),u_(b1), . . . , u_(b(p−1)), u_(a,np)], as well as u_(c) ^(T)=[u_(a,11),. . . , u_(a,(n−1)p), u_(ml1), . . . , u_(ml(p−1))]. The magnetic fluxesand voltages of the tree and of the co-tree are correlated via thematrix V in the form Φ_(t)=VΦ_(c) and u_(c)=−V^(T)u_(t), wherein thematrix V results from the topology of the reluctance network RN_(L) andthe division into tree and co-tree. The matrix V can again be dividedinto V^(T)=[V_(s) ^(T) V_(m) ^(T) V_(g) ^(T)] in accordance with theabove division. The matrix V is dependent on the selected number ofstator tooth elements.

The matrix V_(s) for n_(c) drive coils 4 (and thus n=2n_(c)+1 teeth) canbe represented as V_(s)=[V_(s1), 0, V_(s2), 0, V_(s3), 0, V_(s4), 0,V_(s5), 0, 0, 0, 0], wherein the elements 0 describe zero vectors andV_(sj), j=1, . . . , 5 are lower triangular matrices with the dimensiondim(V_(sj))=(n−1)×(n−1), and all entries of this triangular matrix are1.

For example, the matrix V_(m) for five drive magnets 8 (p=5) results asV_(m)=[V_(m1), V_(m2), V_(m3), V_(m4), 0, . . . , 0, V_(ml)], with thesubmatrices of the dimension dim(V_(mj))=p×n, j=1, . . . , p−1. Thesesub matrices result, for example, as

${V_{m1} = \begin{bmatrix}1 & \cdots & 1 \\0 & \cdots & 0 \\0 & \cdots & 0 \\0 & \cdots & 0 \\{- 1} & \cdots & {- 1}\end{bmatrix}},{V_{m2} = \begin{bmatrix}0 & \cdots & 0 \\{- 1} & \cdots & {- 1} \\0 & \cdots & 0 \\0 & \cdots & 0 \\{- 1} & \cdots & {- 1}\end{bmatrix}},{V_{m3} = \begin{bmatrix}0 & \cdots & 0 \\0 & \cdots & 0 \\1 & \cdots & 1 \\0 & \cdots & 0 \\{- 1} & \cdots & {- 1}\end{bmatrix}},{V_{m4} = \begin{bmatrix}0 & \cdots & 0 \\0 & \cdots & 0 \\0 & \cdots & 0 \\{- 1} & \cdots & {- 1} \\{- 1} & \cdots & {- 1}\end{bmatrix}},{V_{m1} = {\begin{bmatrix}1 & 0 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & {- 1} & 1 & 0 \\0 & 0 & 1 & {- 1} \\0 & 0 & 0 & {- 1}\end{bmatrix}.}}$

From the correlations of the magnetic fluxes and voltages of the treeand the co-tree, the matrix V_(g) also follows, wherein this matrix inturn contains 0, 1.

The magnetic flux Φ and the magnetic voltages u are moreover alsocorrelated via the permeances in the form Φ_(tg)=G_(t)u_(tg) andΦ_(c)=G_(c)u_(c) with the permeance matrices of the treeG_(t)=diag[G_(m) . . . G_(m) G_(b) . . . G_(b) G_(a,np)] and of theco-tree G_(c)=diag[G_(a,11) . . . G_(a,(n−1)p) G_(ml) . . . G_(ml)].

In order to obtain the system equations for the entire reluctancenetwork RN (RN_(s)+RN_(L)), the magnetic flux Φ_(Σ) in the aboveboundary condition at the boundary at y=0 is expressed by means of themagnetic fluxes in the air gap, which results in

−T₀Φ̂ + ? + G_(s0)T_(L)^(φ)T₀φ̂ = 0.?indicates text missing or illegible when filed

The matrices T_(c) ^(Φ) and T_(t) ^(Φ) select the air gap fluxes foreach of the n teeth and sum them. The result is, for example,Φ_(Σ1)=Σ_(jα1) ^(p)Φ_(a,1j).

The above equations which result from the reluctance network of thestator RN_(s) and the reluctance network of the rotor RN_(L) can bemerged, from which the overall system equations of the entire reluctancenetwork RN can be represented in the form of an equation systemK(x)x−B(i_(c))=0. This equation system represents the model of thelinear motor 1 and can be used as constraint g(SG,x) with SG=i_(c) andthe state vector x or for determining a variable of a quality termJT_(k)(SG) of the quality functional J(SG). This representation can bespecified for any topology of the reluctance network RN.

For the exemplary embodiment according to FIGS. 3, 4 and 5 , the statevector with the state variables x^(T)=[{circumflex over (Φ)}^(T){circumflex over (φ)}^(T) u_(tg) ^(T) Ψ_(c) ^(T)] and the overall systemmatrix

${K(x)} = \begin{bmatrix}\overset{\sim}{D} & {- {\overset{\sim}{A}}_{1}} & 0 & 0 \\{- {{\overset{\sim}{A}}_{2}( \hat{\Phi} )}} & \overset{\sim}{D} & 0 & 0 \\T_{0} & {( {{T_{c}^{\Phi}G_{c}V_{s}^{T}T_{s}^{\varphi}} - {G_{s0}T_{L}^{\varphi}}} )T_{0}} & {- ( {\text{?} - {T_{c}^{\Phi}\text{?}V_{g}^{T}}} )} & 0 \\T_{L} & {G_{sy}T_{L}^{\varphi}T_{L}} & 0 & 0 \\0 & {V_{g}G_{c}V_{s}^{T}T_{s}^{\varphi}T_{0}} & {\text{?} + {V_{g}G_{c}V_{g}^{T}}} & 0 \\W & 0 & 0 & \text{?}\end{bmatrix}$ ?indicates text missing or illegible when filed

result. The input vector follows as

${B( \text{?} )} = {\begin{bmatrix}0 \\\text{?} \\{{- \text{?}}V_{m}^{T}\text{?}} \\0 \\{{- V_{g}}G_{c}V_{m}^{T}\text{?}} \\0\end{bmatrix}.}$ ?indicates text missing or illegible when filed

In order to obtain a well-defined equation system, a row [0 0 . . .1]{circumflex over (Φ)}=e_(φ) ^(T){circumflex over (φ)}=0, with which{circumflex over (Φ)}_(n(N+1))=0 is defined, can be added to the overallsystem equations.

All system variables can thus be calculated with the model, inparticular the linked magnetic fluxes Ψ_(c) of the drive coils 4 as afunction of the coil currents i_(c). The overall system equation systemcan also be converted easily so that the drive currents i_(c) aredetermined as a function of the magnetic fluxes Ψ_(c).

In the case of a linear motor, the coil voltages v_(c) are usually usedas an input variable in that the coil voltages v_(c) are generated andapplied to the drive coil by means of the power electronics system. Byusing the known law of induction

${\frac{d\text{?}}{dt} = {{L\frac{d\text{?}}{dt}} = {{- \text{?}} + \text{?}}}},$?indicates text missing or illegible when filed

with the known ohmic resistance R_(c) and the known inductance L of thedrive coils 4, the coil voltage v_(c) can be determined from the coilcurrent i_(c), and vice versa.

From the system equations of the reluctance network RN, the drivingforces and/or driving torques, for example the propulsive force F_(x)and the normal force F_(y) in the form

$F_{x} = {{\frac{1}{2}\text{?}\frac{\partial\text{?}}{\partial\text{?}}\text{?}} + {\frac{1}{2}\text{?}\frac{\partial\text{?}}{\partial\text{?}}\text{?}}}$$F_{y} = {{\frac{1}{2}\text{?}\frac{\partial\text{?}}{\partial\text{?}}\text{?}} + {\frac{1}{2}\text{?}\frac{\partial\text{?}}{\partial\text{?}}{\text{?}.}}}$?indicates text missing or illegible when filed

can also be derived, for example by using the known principle ofmagnetic co-energy. Thus, the driving forces, and also the drivingtorques, acting on the transport unit 3 can also be determined as afunction of the relative position [x_(s), y_(s)] of the transport unit 3relative to the stator 2. The driving forces and driving torques cansupplement the model of the long-stator linear motor as a linear motor1.

The described reluctance model for the long-stator linear motor cannaturally be expanded to the case of a planar motor as a linear motor 1.

1. A method for operating a linear motor comprising a stator, on which aplurality of drive coils is arranged, and comprising a transport unit,which is moved along the stator and on which a plurality of drivemagnets is arranged, wherein an electromagnetic field, which interactswith the drive magnets of the transport unit for moving the transportunit, is generated by energizing active drive coils in the region of thedrive magnets of the transport unit, wherein a quality functional J(SG)is used as a function of manipulated variables (SG) of the active drivecoils, wherein the quality functional J(SG) contains a sum of a numberk≥1 of quality terms JT_(k)(SG), which are weighted with weightingfactors k_(k) and are dependent on the manipulated variables (SG),wherein the quality functional J(SG) contains a quality term JT_(k)(SG),which evaluates a deviation of a specified movement setpoint variable ofthe control of the movement of the transport unit from an actualvariable of the control which is dependent on the manipulated variable(SG), wherein the quality functional J(SG) is optimized for the controlof the movement of the transport unit along the stator with regard tothe manipulated variables (SG) in order to determine optimal manipulatedvariables (SG_(opt)) for the relevant time step of the control of themovement, and the active drive coils are energized according to thedetermined optimal manipulated variables (SG_(opt)), and wherein duringthe movement of the transport unit along the stator, at least twomovement phases are provided, wherein in the at least two movementphases, different quality functionals J(SG) are used for determining theoptimal manipulated variables (SG_(opt)), wherein the different qualityfunctionals J(SG) differ by the number k of the quality terms JT_(k)(SG)used and/or by the quality terms JT_(k)(SG) and/or by the weightingfactors k_(k).
 2. The method according to claim 1, wherein the qualityfunctional J(SG) is optimized on-line during the movement of thetransport unit in each time step of the control in order to determinethe optimal manipulated variables (SG_(opt)) in the relevant time stepof the control of the movement.
 3. The method according to claim 1,wherein the quality functional J(SG) is optimized off-line for aspecified movement of the transport unit along the stator in order tocalculate in advance the optimal manipulated variables (SG_(opt)) to beset during the performance of the movement.
 4. The method according toclaim 1, wherein for the different quality functionals J(SG) of the atleast two movement phases, movement characteristic maps, which map amovement setpoint variable and a position of the transport unit relativeto the active drive coils to optimal manipulated variables (SG_(opt)),are created in advance by off-line optimization, and wherein during theperformance of the movement of the transport unit, the associatedoptimal manipulated variables (SG_(opt)) are read out from a movementcharacteristic map on the basis of a determined actual position of thetransport unit relative to the active drive coils and a specifiedsetpoint value of the movement setpoint variable.
 5. The methodaccording to claim 1, wherein the optimal manipulated variables(SG_(opt)) are adjusted in a manipulated variable controller.
 6. Themethod according to claim 1, wherein the quality functional J(SG)contains a quality term JT_(k)(SG) which evaluates the electrical powerrequired for the movement of the transport unit.
 7. The method accordingto claim 6, wherein as a quality term JT_(k)(SG), the square of theEuclidean norm of a vector with the manipulated variables (SG) of theenergized drive coils is used, i.e., JT_(k)(SG)=k_(i)·∥SG∥₂ ².
 8. Themethod according to claim 1, wherein the quality functional J(SG)contains a quality term JT_(k)(SG) of the formJT_(k)(SG)=k_(e)(BS−IS(SG))² or JT_(k)(SG)=k_(e)(BS−IS(SG)).
 9. Themethod according to claim 1, wherein the quality functional J(SG)contains a quality term JT_(k)(SG) which evaluates a sum SG_(Σ) of themanipulated variables (SG), preferably as a square of the sum, i.e.,JT_(k)(SG)=k_(Σ)SG_(Σ) ² or JT_(k)(SG)=k_(Σ)SG_(Σ).
 10. The methodaccording to claim 1, wherein a quality term JT_(k)(SG) contains avariable which is determined using a mathematical model of the linearmotor.
 11. The method according to claim 1, wherein a constraint g(SG)in the form of a mathematical model of the linear motor is taken intoaccount in the optimization.
 12. The method according to claim 10,wherein the linear motor is modeled with a reluctance network as anetwork with branches, in which reluctances R, permeances G or magneticvoltage sources are arranged and in which magnetic fluxes Φ flow. 13.The method according to claim 12, wherein the stator of the linear motoris modeled with a number n of teeth connected by a stator yoke, whereina differential section dy of each of the number n of teeth is modeled bya branch of the reluctance network of the stator with a differentialreluctance R′_(α)(Φ_(i)), R′_(c)(Φ_(i)) and, in the case of a drivecoil, on one of the number of teeth with a magnetic voltage source witha differential magnetic voltage u′_(c,j), wherein a magnetic potentialφ_(n)(y), φ_(n)(y+dy) is applied to the ends of each branch, and amagnetic flux Φ_(n)(y) flows in a branch and two adjacent branches areconnected by a differential permeance G′, which models a magneticcoupling of two adjacent teeth through a tooth interspace to form nodesof the reluctance network of the stator, wherein the stator yoke ismodeled by linear permeances G_(sy), which connect two adjacent branchesof the reluctance network of the stator (RNs) to form further nodes ofthe reluctance network of the stator (RNs), and wherein an air gap atthe end of the stator opposite the stator yoke is modeled by leakagepermeances G_(s0), which connect two adjacent branches of the reluctancenetwork of the stator (RN_(s)) to form further nodes of the reluctancenetwork of the stator, and by a magnetic flux Φ_(Σn) in the air gap. 14.The method according to claim 13, wherein at each node of the reluctancenetwork of the stator, a node sum of the magnetic fluxes Φ_(n)(u) in thebranches and of the magnetic potential φ_(n)(y), φ_(n)(y+dy) on thebranches is formulated in order to obtain, as system equations of thestator, an equation system that models the stator of the linear motor.15. The method according to claim 12, wherein the transport unit and anair gap between teeth of the stator and the drive magnets on thetransport unit, which are connected by a magnet yoke, are modeled with areluctance network of the rotor, in which the p drive magnets aremodeled as branches of the reluctance network of the rotor with a linearpermeance G_(m) and with a magnetic voltage u_(msp) of a magneticvoltage source, wherein the ends of the branches are connected byleakage permeances G_(ml) at the air gap and linear permeances G_(b) atthe magnet yoke to form nodes of the reluctance network of the rotor,wherein the nodes at the air gap are linked to further branches of thereluctance network of the rotor, in which air gap permeances G_(a,ij)are arranged, wherein each of the p drive magnets in the reluctancenetwork of the rotor is connected via a respective air gap permeanceG_(a,ij) to each of the n teeth in the reluctance network of the stator,and wherein the branches with the air gap permeances G_(a,ij) areconnected by voltage sources representing the stator to the magneticvoltages u_(s) to respectively form further nodes of the reluctancenetwork of the rotor.
 16. The method according to claim 15, wherein fromthe reluctance network of the rotor, an equation system which models theair gap and the transport unit is formed as system equations of therotor.
 17. The method according to claim 14, wherein from the systemequations formed from the reluctance networks, driving forces and/ordriving torques acting on the transport unit are derived.
 11. A linearmotor comprising a stator, on which a plurality of drive coils isarranged, and comprising a transport unit, which is moved along thestator and on which a plurality of drive magnets is arranged, whereinactive drive coils in the region of the drive magnets of the transportunit are energized in order to generate an electromagnetic field,wherein the electromagnetic field interacts with the drive magnets ofthe transport unit for moving the transport unit, wherein a control unitis provided, in which a quality functional J(SG) is implemented as afunction of manipulated variables (SG) of the energized active drivecoils, wherein the quality functional J(SG) contains a sum of a numberk≥1 of quality terms JT_(k)(SG), which are weighted with weightingfactors k_(k) and are dependent on the manipulated variables (SG),wherein the quality functional J(SG) contains a quality term JT_(k)(SG),which evaluates a deviation of a specified movement setpoint variable ofthe control of the movement of the transport unit from an actualvariable of the control which is dependent on the manipulated variable(SG), wherein the control unit optimizes the quality functional J(SG)for the control of the movement of the transport unit along the statorwith regard to the manipulated variables (SG) in order to determineoptimal manipulated variables (SG_(opt)) for the relevant time step ofthe control of the movement, and the drive coils are energized accordingto the determined optimal manipulated variables (SG_(opt)), and whereinduring the movement of the transport unit along the stator, at least twomovement phases are provided in the control unit, wherein differentquality functionals J(SG) for determining the optimal manipulatedvariables (SG_(opt)) are implemented for the at least two movementphases, wherein the different quality functionals J(SG) differ by thenumber k of the quality terms JT_(k)(SG) used and/or by the qualityterms JT_(k)(SG) and/or by the weighting factors k_(k).